# A Survey of Lie Groups and Lie Algebra with Applications and by Johan G. F. Belinfante

By Johan G. F. Belinfante

During this reprint version, the nature of the booklet, in particular its concentrate on classical illustration conception and its computational elements, has no longer been replaced

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**Example text**

14 Lie Subgroups and Homomorphisms 31 The creation and annihilation operators satisfy the commutation relations with each other, and with the Hamiltonian. These latter commutation relations imply that if ij/ is an eigenvector of H with eigenvalue E, then ax\}/ is either zero or an eigenvector with eigenvalue E — 1, while a* if/ is an eigenvector with eigenvalue E + 1. Thus the harmonic oscillator has an infinite series of equally spaced levels. Both the Hamiltonian and the commutation relations are invariant under a unitary transformation, so U(n) is a symmetry group of the n-dimensional harmonic oscillator.

Conversely, a complex Lie algebra L may be regarded as a real Lie algebra LR, since the multiplication of Lie elements by real numbers may be defined simply as the obvious restriction of the multiplication by complex numbers. This Lie algebra LR is called the real restriction of the complex Lie algebra L. Thus we can pass fro;n real Lie algebras to complex Lie algebras, and back again. 20 DIRECT SUMS OF VECTOR SPACES The direct sum, like the tensor product, is a fundamental vector space operation which finds many applications in the theory of Lie algebras and their representations.

If we take X, to be the linear operator corresponding to multiplication of any function ij/(\) by jc,-, and P{ to be the differential operator —ihd/dx^ then a purely formal computation indicates that these commutation relations are fulfilled. In the Schrodinger formulation of nonrelativistic quantum mechanics, these operators are interpreted as Hermitian operators on an appropriate Hilbert space of equivalence classes of square-integrable functions. The quantization prescription given above is unfortunately not complete.